Basic Computer Math
Conversion of Numbers of Different Bases to Decimals
Consider the number 97531.
This is really: 9×10000 + 7×1000 + 5×100 + 3×10 + 1.
This can also be written as: 9×104 + 7×103 + 5×102 + 3×101 + 1×100.
There is nothing special about why we use base 10 except that humans
have ten fingers.
In the world of electronics, the natural base is two, because it only has two
states: on and off. We use “1” to signify on, and “0” to signify off. This is
referred to as the “binary system.”
Example 1: Consider the binary number 1101011001.
To convert this to its decimal equivalent, we rewrite it as:
1×29 + 1×28 + 0×27 + 1×26 + 0× 25 + 1×24 + 1×23 + 0×22 + 0×21 + 1×20.
Or:
1×512 +1× 256 + 0×128 + 1×64 + 0×32 + 1×16 + 1×8 + 0×4 + 0×2 + 1 = 857.
Computational Devise “A”
Powers of 2
Binary Digits
Their Product
1024
512
256
128
64
32
16
8
4
2
1
1
512
1
256
0
0
1
64
0
0
1
16
1
8
0
0
0
0
1
1
Their sum: 512 + 256 + 0 + 64 + 0 + 16 + 8 + 0 + 0 + 1 = 857
Suppose we were to use another base, say “b.” That would mean that the
numbers representing the amounts multiplying their powers would range
from “0” to one less that the base, i.e. “b-1.” For example, in base ten,
these numbers range from “0” to “9.” That is why in base two we only
have “0” and “1.” The commonly used base in Information science is base
sixteen, or hexadecimal (“hex” means six and “deci” refers to tenth).
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Example 2: Consider the base four number (1023)4. Notice the numbers
range between “0” and “3,” the subscript “4” indicates the base of the
number.
1×43 0×42 + 2×41 + 3×40
1×64 + 0×16 + 2×4 + 3×1 = 73
Powers of 4
256
Base 4 Digits
Their Product
64
16
4
1
1
64
0
0
2
8
3
3
Adding the digits in the last row: 64 + 0 + 8 + 1 = 73
Example 3: (647)8
6×82 + 4×81 + 7×80
6×64 + 4×8 + 7×1 = 415
Powers of 8
512
Octal Digits
Their Product
64
8
1
6
384
4
24
7
7
The sum of the digits in the last row 384 + 24 + 7 = 415
Notice that we have chosen bases that are powers of two. Those are the
numbers that followed the progression within computer science. Because
it is more convenient to use higher bases to represent quantities because it
requires less space, or slots, to do so. The final base that the
computational progression settled on was base sixteen or hexadecimal.
Using base16, the range of numbers multiplying the powers of the base
have to range between “0” and “fifteen.” However, it was decided to
represent the amounts from “ten” to “fifteen” by the letter A, B, C, D, E and
F (i.e. A=10, B=11, C=12, D=13, C=14 and F=15).
Example 4: (D3)16
D×161 + 3×160
D×16 + 3×1 = 13×16 + 3×1 = 209
Most of the hexadecimal numbers used in computer math will only have
two places, thus representing numbers from 0 to FF (or 0 to 255).
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Conversion of Decimals Numbers to Different Bases
Consider the number 97531.
Computational Devise “B”
Base Divisor
10
10
10
10
10
Dividend/Quotient
97531
9753
975
97
9
0
Remainder
1
3
5
7
9
Check:
Powers of 1
100000
Decimal Digits
Their Product
10000
1000
100
10
1
9
90000
7
7000
5
500
3
30
1
1
The sum of the digits in the last row is:
90000 + 7000 + 500 + 30 +1 = 97531
Notice how when you divide each dividend the remainder is the quantity
that multiplies the base to the next lower power. And when you have a
quotient of “0” you have finished, and the final remainder is the quantity
that multiplies the highest powered base.
Example 5: Convert the decimal number 25 to its binary equivalent.
Base Divisor
2
2
2
2
2
Dividend/Quotient
25
12
6
3
1
0
3
Remainder
1
0
0
1
1
Check:
Powers of 2
1024
512
256
128
64
32
Binary Digits
Their Product
16
8
4
2
1
1
16
1
8
0
0
1
0
1
1
The sum of the digits in the last row 16×1 + 1×8 + 0×4 + 1×2 + 1×1 = 25
Example 6: (48307)16
Base Divisor
16
16
16
16
16
Dividend/Quotient
48307
3019
188
11
0
Remainder
3
11
12 = C
11 = B
Check:
Powers of 16
4096
256
16
1
Hexadecimal Digits
Their Product
B = 11
45056
C = 12
3072
11
176
3
3
Their sum 45056 + 3072 + 176 +3 = 48307
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Hexadecimal Equivalents to the Decimal Numbers 0–255
Dec
Hex
Dec
Hex
Dec
Hex
Dec
Hex
Dec
Hex
Dec
Hex
Dec
Hex
Dec
Hex
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
10
11
12
13
14
15
16
17
18
19
1A
1B
1C
1D
1E
1F
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
20
21
22
23
24
25
26
27
28
29
2A
2B
2C
2D
2E
2F
30
31
32
33
34
35
36
37
38
39
3A
3B
3C
3D
3E
3F
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
40
41
42
43
44
45
46
47
48
49
4A
4B
4C
4D
4E
4F
50
51
52
53
54
55
56
57
58
59
5A
5B
5C
5D
5E
5F
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
60
61
62
63
64
65
66
67
68
69
6A
6B
6C
6D
6E
6F
70
71
72
73
74
75
76
77
78
79
7A
7B
7C
7D
7E
7F
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
80
81
82
83
84
85
86
87
88
89
8A
8B
8C
8D
8E
8F
90
91
92
93
94
95
96
97
98
99
9A
9B
9C
9D
9E
9F
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
A0
A1
A2
A3
A4
A5
A6
A7
A8
A9
AA
AB
AC
AD
AE
AF
B0
B1
B2
B3
B4
B5
B6
B7
B8
B9
BA
BB
BC
BD
BE
BF
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
C0
C1
C2
C3
C4
C5
C6
C7
C8
C9
CA
CB
CC
CD
CE
CF
D0
D1
D2
D3
D4
D5
D6
D7
D8
D9
DA
DB
DC
DD
DE
DF
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
E0
E1
E3
E3
E4
E5
E6
E7
E8
E9
EA
EB
EC
ED
EE
EF
F0
F1
F2
F3
F4
F5
F6
F7
F8
F9
FA
FB
FC
FD
FE
FF
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